Impossible quest: integrate sqrt(x)*cosx dx?

Answer 1

If you mean: Find the primitive function of sqrt(x)*cos(x) in terms of elementary functions, then this truly can not be done.

On the other hand, the integrating function is continuous on every segment for x>0, so it is Riemann integrabale. You can find the solution in terms of infinite series. Namely:

cos(x) = Series[(-1)^n*x^(2n)/(2n)!,{n,0,Infinity}]

so sqrt(x)*cos(x) = Series[(-1)^n*x^(2n+1/2)/(2n)!,{n,0,Infinity}]

Integrating, we get:

Integral = C + Series[(-1)^n*x^(2n+3/2)/((2n)!*(2n+3/2)),{n,0,Infinity}]

It can be expressed in terms of Fresnel's Integrals!

Answer 2

Integrate (√x)(cosx).

This is a Fresnel integral and well beyond first or second year calculus. Below is a link to the Wolfram Integrator. Just copy and paste the following:

x^(1/2) Cos[x]

http://integrals.wolfram.com/index.jsp

Answer 3

suppose that sqrt(x)=t
then
squaring on both sides
we get
x=t^2 dx=2tdt
substitute it
integrate t*cost^2*2tdt =2t^2*cost^2dt

by method of integration by parts

=2t^2*integralcost^2 dt -integral[derivative of t^2*integralcost^2dt]

2t^2sint^2/2t -integral[4t*sint^2/2t]

=t*sint^2 -integral[2*sint^2]
=t*sint^2 +2*cost^2/2t +constant
but x=t^2 =>t=rootx
=rootx*sinx +cosx/rootx +constant

Answer 4

sqrt(x)*sinx -sqrt(pi/2) S(sqrt(2x/pi)) + constant

where S is a Fresnel integral.

Answer 5

If you meant intergral of (xcosx)^(1/2)
then the integral is: x(xcosx)^(1/2)